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Why  Do  We  Study  Mathematics: 
A  Philosophical  and  Historical  Retrospect 


By  Thomas  J.  McGtrmack 


Why  Do  We  Study  Mathematics: 
A  Philosophical  and  Historical  Retrospect 


Address  Delivered  Before  the  Secondary  Mathematics 

Section  of  the  National  Education  Association 

Boston,  July  8,  1910 


By  Thomas  J.  McCormack 

Principal  of  the  La  Salle-Peru  Township  High  School 
La  Salle,  Illinois 


THE  TORCH  PRESS 

CEDAR  RAPIDS,  IOWA 

1910 


a- 


v\ 


'  •  i .'  • 


WHY  DO  WE  STUDY  MATHExM  ATICS : 
A  PHILOSOPHICAL  AND  HISTORICAL  RETROSPECT 

Introduction 

The  thought  that  impresses  itself  most  forcibly  upon  one's 
mind  as  one  approaches  this  subject  of  the  part  which  the 
study  of  mathematics  plays  in  education,  is  the  extremely 
fortunate  and  exalted  position  that  this  discipline  occupies 
in  the  hierarchy  of  the  sciences  that  have  been  selected  as 
especially  designed  to  inform  and  cultivate  the  human  mind. 
The  oldest  branch  of  human  knowledge  to  be  investigated, 
the  first  to  take  systematic  and  dogmatic  form,  it  has,  in  the 
three  thousand  or  more  years  of  its  development,  rarely  halted 
in  its  advances  and  never  ceased  to  fructify  either  the  fields 
of  practical  knowledge  or  the  loftier  realms  of  metaphysical 
thought.  Men  of  affairs  and  philosophers,  whether  for  ma- 
terial gain  or  pure  intellectual  enhancement,  have,  for  ages, 
alike  looked  to  it  for  succor  and  guidance;  and  wherever,  in 
the  history  of  human  thought,  it  has  appeared  that  the  ideals 
of  truth  and  certitude  were  destined  to  extinction,  there  in 
its  despair  thinking  humanity  has  found  in  the  irrefragable 
conclusions  of  mathematics  an  unfailing  solace  and  support. 
Encircled  with  a  halo  of  divinity  by  Pythagoras  and  Plato, 
with  whom  the  laws  of  _naiura-ware  but-  the  geometricaL 


thoughts  oF"(jod,  it  ran  serene  and  undisturbed  the  solitary 
gamut  of  its  development  through  the  entire  history  of  phil- 
osophy, embracing,  dominating,  and  even  submerging  in  Des- 
cartes, Spinoza,  and  Kant  the  whole  structure  of  human 
knowledge," and  furnishing  the  prototype  of  all  human  inves- 
tigation of  truth.  And  in  the  great  scientific  compeers  of 
these  last-named  men,  Galileo,  Huygens,  Newton,  Euler,  La- 
place, Monge,  and  Lagrange,  it  took  that  direct  practical  turn 
that  made  it  the  living  groundwork  of  all  the  marvels  of  the 


233109 


material  civilization  of  whicli  we  6t  today  boast.  On  neither 
its  theoretical  nor  its  utilitarian  side,  therefore,  does  it  need 
apology  or  justification.  It  needs  simply,  for  our  present  pur- 
poses, analysis;  and  this,  under  one  or  two  simple  points  of 
view,  without  any  pretence  of  exhausting  a  well-worn  subject, 
I  purpose  to  offer,  in  the  hope  that  the  clarification  which  the 
process  involves  will  present  some  important  pedagogical  im- 
plications. 

MATHEMATICS   AN    ECONOMIZATION   OF   THOUGHT,    A    CAPITALIZA- 
TION OF  INTELLECTUAL  LABOR-SAVING  DEVICES 

The  most  salient  practical  feature  of  all  scientific  thought 
is  its  economic  or  labor-saving  purpose.  Even  our  every-day 
modes  of  thinking  are  impregnated  with  this  trait.  Education 
itself  largely  consists  of  the  inculcating  of  ready-made,  snap 
judgments  on  men  and  events,  on  sociology  and  economics,  on 
politics  and  history.  Prejudice  is  the  supreme  type  of  ready 
intellectual  power;  the  mightiest  weapon  in  our  logical  arm- 
ory :  it  is  mental  preparedness. 

#  The  laws  of  physics,  of  astronomy,  of  geology,  and  the  rest 
are,  on  their  practical  side,  mere  shorthand  or  rather  short- 
mind  formulae  or  rules  for  recovering,  with  a  minimum  of 
mental  labor,  by  means  of  little  brains  and  the  mechanical 
manipulations  of  a  pencil,  the  past  and  future  facts  of  nature, 
which,  without  these  rules  and  laws,  we  should  have  indefinite 
labor  and  take  indefinite  time  in  recovering.^  Such  are  the 
so-called  physical  laws  of  refraction  and  of  falling  bodies,  the 
prediction  of  eclipses,  etc.  Millions  of  cases  reduced  to  a 
single  case.  'All  a  saving  of  intellectual  labor. 

But  this  economy  of  mental  effort  is  most  conspicuous  in 
mathematics.  Memorized  addition-results  save  not  only  fric- 
tion of  aboriginal  fingers  and  toes,  but  save  also  brain-strain, — 

1  For  the  full  development  of  this  view  of  science  as  an  economy 
of  thought  see  the  works  of  Ernst  Mach,  especially  the  "Science  of 
Mechanics"  and  ** Popular  Scientific  Lectures,"  translated  by  T.  J. 
McCormack  and  published  by  the  Open  Court  Publishing  Co.,  Chicago. 
The  idea  was  contained  in  nuoe  in  the  works  of  Adam  Smith,  Babbage, 
and  several  German  writers  quoted  by  Mach.  It  has  also  found  sporadic 
expression  in  such  quotations  as  that  given  in  this  section  from  De 
Morgan. 


an  economy  still  further  heightened  and  typified  by  the  mod- 
ern adding-machines.  The  multiplication-table,  the  Arabic 
decimal  machinery,  determinants,  integration  are  all  thought- 
saving  devices.  But  preeminent  among  all  these  economic 
mechanizations  of  human  thinking  are  the  tables  of  logarithms 
and  products,  powers,  roots,  interest  and  annuity  tables,  etc., 
in  which  the  mathematical  labors  of  generations  have  been 
capitalized  and  amassed  for  all  time. 

In  this  view  the  development  of  mathematics  becomes  a 
continuously  progressive  abstract  economization  and  petrifac- 
tion of  quantitative  thought,  a  permanent  crystalization  of 
quantitative  architectonic  thinking,  a  standardizing  of  the  ma- 
chinery of  the  mind,  which  reaches  the  pinnacle  of  its  refine- 
ment in  the  shorthand  symbolism,  or,  as  I  prefer  to  call  it,  the 
shortmind  or  stenophrenic  ^  (as  distinguished  from  sten- 
ographic) symbolism  of  algebraic  analysis.  Algebra,  in  this 
view,  to  quote  the  lucid  words  of  an  old  editor  of  Euclid,  is 
that  ''paradise  of  the  mind,  where  it  may  enjoy  the  fruits  of 
all  its  former  labors,  without  the  fatigue  of  thinking. "  ^  It  is 
the  essential,  paradoxical  purpose  of  mathematics,  in  this  con- 
ception, to  get  rid  of  thinking  by  very  dint  of  thinking. 
«  This  is  the  purely  intellectual  side.  But  this  economization 
of  mental  and  manual  labor  can  be  even  more  palpably  traced 
in  the  domain  of  applied  mathematics,  in  engineering,  survey- 
ing, navigation  and  the  rest,  to  develop  which  I  refrain,  from 
lack  of  time. 

PEDAGOGIC  IMPLICATIONS  OF  THIS  VIEV^ 

Now  what  is  the  lesson  of  all  this  ?  I  see  in  it  two  things. 
First,  we  have,  in  this  process,  in  its  purest  and  simplest 
form,  the  original  and  primordial  type  of  all  human  intel- 
lectual activity,  the  incarnate  essence  of  all  human  mental 
striving,  which  is  intense  economization,  and  progressive  con- 
centration of  mental  effort.  The  student  who  has  gained  this 
point  of  view,  by  contact  with  mathematical  and  especially 
algebraic  study,  acquires  from  it  a  sense  of  intellectual  power 

1  From  ffTev6i,    short,  and  0/ji}v,  4>p€v6s^    mind. 

1  Quoted  by  De  Morgan  in  the  Preface  to  his  * '  Essay  on  Probabil- 
ities," London,  1838. 


and  discipline,  which  can  be  brought  home  to  him  with  equal 
force  by  the  study  of  no  other  branch  of  knowledge,  and  draws 
from  it  an  aesthetic  inspiration  that  heightens  his  whole  spirit- 
ual life.  There  is  a  moral  sense  of  enlightenment  in  it  that 
courts  comparison  with  that  of  any  other  discipline;  and  I 
take  it,  that  this  view  has,  for  its  practical  educational  value, 
not  yet  received  its  proper  emphasis  in  pedagogical  theory. 

And  the  second  lesson  is,  that  this  view,  on  its  ultra-prac- 
tical side,  connects  our  so-called  formal  theoretical  study  by 
a  living  link  with  our  material  industrial  civilization,  whicli 
sees  the  goal  of  life  in  the  progressive  minimization  of  all 
human  labor  and  the  saving  of  human  resources  for  higher 
ends.  Both  the  practical  and  ethical  implications  of  this 
view  are  immediately  apparent,  and  it  is  gratifying  to  see  its 
main  trend  faintly  emphasized  in  some  of  our  recent  text- 
books. No  one  who  can  grasp  this  point  of  view  will  ever 
have  the  hardihood  to  say  that  the  science  of  mathematics  is 
without  a  moral,  let  alone,  an  intellectually  material,  con- 
tent, —  a  moral  content  for  which  alone  it  deserves  to  be 
studied,  entirely  apart  from  its  applications,  which  so  few 
students  ever  make,  and  entirely  apart  from  the  Jogical  cul- 
ture which  it  gives  and  which  has  been  its  glory  for  centuries. 
This  is  one  of  the  great  benefits  derived  from  mathematical 
study,  which  every  thinking  person  will  admit,  which  has  not, 
in  my  opinion,  been  sufficiently  recognized,  but  which  alone 
would  justify  its  pursuit  as  a  branch  of  practical  intellectual 
culture. 

PRACTICAL  INTELLECTUAL  CULTURE 

I  would  emphasize  particularly  this  phrase  I  have  used,  — 
practical  intellectual  culture.  There  are  things  intellectually 
and  spiritually  practical,  as  well  as  things  materially  practical. 
And  the  former  have  as  much  right  to  a  place  in  education, 
from  which  they  would  seem  to  be  now  becoming  displaced, 
as  the  former.  And  unless  we  recognize  this,  one-half  of  the 
studies  of  our  present  secondary  curricula  must  go,  and  espe- 
cially mathematics  must  go;  for  upon  its  meagre  quota  of 
possible  so-called  practical,  material  applications,  for  the  ma- 
jority of  secondary  students,  it  cannot  rest. 


De  Morgan,  as  early  as  1838,  in  the  preface  to  his  "Essay 
on  Probabilities,"  where  he  applied  mathematics  in  the  most 
practical  possible  manner  to  the  questions  of  life-insurance 
and  its  related  subjects,  spoke  with  regret  of  the  tendency 
which  even  in  his  day  existed  both  in  England  and  America 
with  regard  to  opinion  upon  the  end  and  use  of  knowledge, 
and  the  purpose  of  education;  and  his  remarks  have  far  great- 
er applicability  now.  He  says:  ''Of  the  thousands  who,  in 
each  year,  take  their  station  in  the  different  parts  of  busy 
life,  by  far  the  greater  number  have  never  known  real  mental 
exertion;  and,  in  spite  of  the  variety  of  subjects  which  are 
crowding  upon  each  other  in  the  daily  business  of  our  ele- 
mentary schools,  a  low  standard  of  utility  is  gaining  ground 
with  the  increase  of  the  quantity  of  instruction,  which  deteri  • 
orates  its  quality.  All  information  begins  to  be  tested  by  its 
professional  value;  and  the  knowledge  which  is  to  open  the 
mind  of  fourteen  years  old  is  decided  upon  by  its  fitness  to 
manure  the  money-tree." 

«       FORM   OR  CONTENT 

But  this  leads  to  one  of  the  celebrated  moot  points  of  edu- 
cational theory,  —  whether  mathematics  should  be  studied  for 
its  intellectual  or  for  its  material  content.  And  a  glance  at 
the  history  of  our  subject  may  throw  some  light  on  this  con- 
troversy. Personally,  I  believe  that  both  demands  can  be 
simultaneously  satisfied ;  but  the  proper  sphere  of  each  should 
first  be  clearly  exhibited  by  analysis,  and  the  rights  and  claims 
of  each  recognized  and  placed  in  their  proper  setting.  I  am 
the  advocate  of  neither  view;  I  am  a  believer  in  both.  The 
truth  is  more  important  than  either ;  and  the  truth  will  make 
us  free  to  teach  according  to  reasoned  and  just  convictions. 

KANT  vs.   SCHOPENHAUER 

The  dominating  and  exclusive  position,  above  referred  to, 
which  mathematics  has  occupied  in  the  hierarchy  of  the  sci- 
ences since  Plato,  whose  utterances  on  its  importance  as  a 
cultural  discipline  are  now  the  commonplaces  of  pedagogy,  has 
led  not  only  to  the  sublimest  flights  of  speculative  thought 
and  a  consequent  apotheosis  of  the  value  of  mathematical 

7 


V 


training,  but  also  to  the  most  extravagant  hyper-emphasis  of 
mathematics  as  an  engine  of  scientific  and  philosophic  enquiry. 
Descartes  saw  truth  only  in  geometry.  Newton,  by  his  en- 
thronement of  the  deductive  method,  retarded  the  advance  of 
English  science  for  a  century.  *And  Kant  went  so  far  as  to 
say  that  "in  every  branch  of  natural  knowledge  there  is  just 
so  much  and  only  so  much  genuine  science  as  there  can  be 
mathematics  applied  in  it. "  ^  The  world  went  mathematics 
mad.  The  calculus  was  as  fashionable  a  fad  in  the  18th 
century  almost  as  Christian  Science  is  today.  Ladies  at  Ver- 
sailles and  the  minor  courts  of  Germany  juggled  with  differ- 
ential coefficients  and  discussed  the  precession  of  the  equinoxes 
at  their  toilettes,  while  Euler's  letters  on  physics  and  deduc- 
tive logic  to  the  Princess  Elizabeth  rivalled  in  continental  pop- 
ularity the  epistles  of  Lord  Chesterfield.  There  were  even 
mathelnatical  schools  of  agriculture,  and  a  mathematical  school 
of  forestry  lasted  in  Germany  even  into  the  19th  century. 
The  influences  of  the  movement  can  be  seen  even  in  some 
modern  phases  of  the  science  of  psychology. 

The  reaction  was  imperative  and  it  was  trenchant.  There 
is  a  proverb  of  unknown  origin  to  the  effect  that  ''purus 
matliematicus,  purus  asinuSy''  or  **Show  me  a  pure  mathema- 
tician and  I  will  show  you  a  simon-pure  jackass."  Frederick 
the  Great  called  his  mathematical  colleagues  of  the  Berlin 
Academy  savages,  hollow-pates  and  lack- wits.-  Napoleon  said 
that  Laplace  *  *  carried  the  spirit  of  his  infinitesimal  pettinesses 
into  the  business  of  state, ' '  and  Schopenhauer  bluntly  assever- 
ated that  "where  mathematics  begins,  the  comprehension  of 
phenomena  ceases. "  ^ 

Such  utterances  I  could  quote  ad  libitum;  as  I  could  also 
their  opposites. 

THE  PLATONIC  AND  THE  GASTRONOMIC  POINTS  OF  VIEW 

How,  now,  for  the  purposes  of  practical  pedagogy,  —  and 

1  *  *  Metaphysische  Anf  angsgriinde  der  Naturwissenschaf  t. ' '     Vorwort. 

2''0euvres  de  Frederic  le  Grand,''  tome  9  (6d.  Decker,  1848),  pp. 
63-65,  68,  72-74;  tome  19  (ed.  1852),  p.  321;  t.  22,  p.  181;  p.  199;  t.  23, 
pp.  417-421. 

3  ' '  Parerga  und  Paralipomena, ' '  Vol.  II.     §  80. 

8 


this  is  our  purpose  here,  —  are  these  strident  contradictions 
to  be  reconciled?  Why  was  it  that  Plato  could  say,  as  he 
did  say  in  his  ' '  Laws, ' '  ^  that  the  man  who  did  not  understand 
the  theory  of  incommensurables  was  no  better  than  a  pig,  that 
question-games  on  the  ratios  of  incommensurable  numbers 
were  a  far  more  graceful  way  of  passing  the  time  for  educated 
men  than  playing  checkers;  while  Savarin,  the  great  French 
gastronomer,  could  proclaim,  amid  the  applause  of  his  con- 
temporaries, when  Leverrier  discovered  by  pure  deductive 
mathematics,  Neptune,  that  ''the  cook  who  invented  a  new 
dish  was  a  greater  benefactor  of  humanity  than  the  savant 
who  discovered  a  new  planet ! ' ' 

Here  is  trenchantly  expressed  the  gaping  hiatus  that  di- 
vides the  camps  of  constructive  educationists  with  regard  to 
the  value  and  the  aims  of  mathematical  study.  I  would  call 
these  two  points  of  view  (1)  the  Platonic  point  of  view,  and 
(2)  the  gastronomic  point  of  view ;  the  point  of  view  of  pure  in- 
tellectual culture,  and  the  point  of  view  of  pure,  practical, 
bread-and-butter  utility.  Which  is  to  be  predominant  in  the 
emphasis  of  topics  and  methods  for  secondary  instruction  in 
mathematics?  Or,  are  the  two  reconcilable;  and  can  both 
aims  be  adequately  satisfied  in  our  instruction  ?  I  believe  they 
can ;  and  that  much  of  our  worry  and  solicitude  on  this  head 
is  misplaced  and  supererogatory;  that  unconsciously  and  un- 
wittingly we  always,  despite  ourselves,  are  fulfilling  both  pur- 
poses. 

^^"^  THE  IDEA  ** practical'*  EXTENDED 

' '  Practical  ' '  and  ' '  utilitarian  ' '  are  current  words  of  pop-  v 
ular  vogue  that  need  sadly  clarification  and  precise  definition. 
Are  not  some  things  '/practical"  and  more  eminently  useful 
for  purely  intellectual,  ethical  and  social  purposes  than  are 
some  other  things  that  find  ready  and  immediate  application 
in  the  measurement  of  concrete  realities^  I  think  we  must 
seek  a  deeper  meaning  for  the  idea  "practical"  here.  Not 
one-tenth  of  the  graduates  of  our  high  schools  ever  enter  pro- 
^  fessions  in  which  their  algebra  and  geometry  are  applied  to 
concrete  realities ;  not  one  day  in  three  hundred  and  sixty-five 

1  Plato,  ''De  Legibus,^'  VII. 


is  a  high  school  graduate  called  upon  to  ''apply,"  as  it  is 
called,  an  algebraic  or  a  geometrical  proposition.  Not  one  in 
ten  of  our  high  school  students  ever  retain  sufficient  mathemat- 
ical knowledge  or  skill  to  solve  even  a  tolerably  difficult  con- 
crete physical  or  mensurational  problem  after  graduation. 
Why  then  do  we  teach  these  subjects,  if  this  alone  is  the  sense 
of  the  word  "practical?"  For  abandon  them,  we  are  all 
agreed  we  should  not. 

To  me  the  solution  of  this  paradox  consists  in  boldly  con- 
fronting the  dilemma  and  in  saying  that  our  conception  of  the 
practical  utility  of  these  studies  must  be  readjusted,  and  that 
we  have  frankly  to  face  the  truth  that  the  "practical"  ends 
we  seek  are  in  a  sense  ideal  practical  ends,  yet  such  as  have 
after  all  an  eminently  utilitarian  value  in  the  intellectual 
sphere.  What  practical  profit  can  a  girl,  or  nine  out  of  ten 
high  school  boys,  derive  from  a  year  and  a  half  of  algebra  — 

-  unless  it  is  the  aesthetic  joy  and  logical  culture  derived  from 
participation  in  the  upbuilding  of  a  great  abstract  structure 
of  symbolic  quantitative  thought,  the  sharing  in  the  dramatic 
triumph  of  a  great  human  intellectual  conquest  (which  I 
referred  to  above  in  my  references  to  mathematics  as  an  econ- 
omization  of  human  thought),  or  that  invaluable,  incalculably 
important  mental  and  manual  drill  (which  Chrystal  has  em- 

^p»hasized),  in  accuracy,  neatness  and  method  that  comes  from 
the  correct  systematic  manipulation  of  algebraic  forms;  or, 
can  the  same  average  student  derive  from  geometry,  save  that 
magnificent  training  in  logical  deductive  power,  in  capacity 
for  intense  concentration  of  the  attention,  in  systematic  un- 
erring pursuit  of  a  goal,  in  intellectual  self-restraint,  and, 
last  but  not  least,  that  intensifying  and  heightening  of  the 

\  power  of  precise  English  expression  and  English  thinking 
that  makes  mathematics  in  our  schools  almost  the  substitute 
and  the  surrogate  of  a  thorough  legal  training! 

Bent  as  I  am  personally  and  naturally  to  the  making  of  all 
mathematical  instruction  practical  and  applied  in  its  most 
popular  sense,  I  must  say  that  in  summing  up  to  myself  the 
results  of  my  own  limited  instruction  I  have  always  felt  that 
the  objects  above  enumerated  have  been  attained  in  my  classes^, 

10 


in  tenfold  greater  degree  than  have  the  so-called  concrete  re- 
sults I  have  so  ardently  sought  and  wished  for,  but  obtained 
only  in  a  limited  number  of  individual  cases. 

We  sometimes  forget  in  our  inordinate  zeal  to  ''practical- 
ize"  and  popularize  education,  that  our  object  is  also  to  make 
men  and  women  as  well  as  engineers  and  "rope-stretchers," 
and  that  the  former  end  is  more  commonly  attained  than  the 
latter.  Our  trouble  and  indecision  arise  from  the  very  im- 
portant psychological  truth  that  it  is  impossible  to  weigh  and 
••measure  psychical  and  ethical  values,  and  that  we  have  not 
always  necessarily  failed  when  we  cannot  palpably  catalogue 
>  our  results.  Education  is  a  subtle  process,  and  withdraws 
itself  from  quantitative  observation.  This  is  why  Plato,  with 
his  divine  tact,  proposed  for  educated  men  the  game  of  tit- 
tat-to  with  incommensurables,  to  replace  the  checkers  of  the 
porklings.  It  is  the  victory  of  the  Platonic  over  the  gastro- 
nomic point  of  view,  the  victory  of  Pythagoras  of  Croton  over 
Mr.  Crane  of  Chicago,  and  leads  us  to  say  with  St.  Paul  that 
for  the  purposes  of  genuine  education  we  ought  sometimes  to 
' '  look  not  at  the  things  which  are  seen,  but  at  the  things  which 
are  not  seen:  for  the  things  which  are  seen  are  temporal;  but 
the  things  which  are  not  seen  are  eternal. "  It  is  these  eternal 
impalpable  things,  which  remain  with  us  all  from  our  study 
of  algebra  and  geometry,  that  constitute  the  sole  profit  that 
ninety  per  cent  of  us  ever  derive  from  the  study  of  mathemat- 
ics, and  it  is  this  that  furnishes  the  foundation  of  truth  to 
Emerson's  paradoxical  saying  that  "education  is  what  remains 
to  us  after  everything  we  have  learned  at  school  is  forgotten. ' ' 

A   CONCRETE   ILLUSTRATION 

I  will  introduce  here  one  concrete  illustration  at  length, 
taken  from  a  London  Conference  of  Secondary  Mathematics 
and  Science  teachers  of  last  year,  as  exemplifying  what  I  mean 
when  I  say  with  Chrystal  that  training  in  habits  of  accuracy, 
mental  and  manual,  is  the  cardinal  benefit  derived  from  the 
study  of  secondary  algebra  for  ninety  per  cent  of  us. 

The  speaker,  Mr.  Jackson,  asks :  ' '  What,  really,  is  our  ob- 
ject in  education;  and  why  is  elementary  science  going  to  do 
all  that  is  expected  of  it?     Mr.  Searle,  in  his  book  on  experi- 

11 


mental  elasticity  just  published,  observes:  'A  demonstrator 
in  physics  spends  much  of  his  time  in  correcting  student's 
mistakes.  He  has  to  discover,  for  instance,  why  a  student  ob- 
tains 537.86402  (no  units  stated)  for  Young's  modulus  for  a 
brass  wire.  .  .  .  The  student  has  confused  radius  with 
diameter;  has  used  a  screw  gauge  in  which  one  turn  is  equiv- 
alent to  1/50  of  an  inch,  and  treated  one  turn  as  equivalent 
to  .5  of  a  millimetre.  .  .  .  has  treated  the  millimetres  as 
if  they  were  centimetres.  .  .  .  and  has  used  32  for  'grav- 
ity' instead  of  981.'  ^ There  is  the  enemy!  The  real  enemy  we 
have  to  fight  against,  whatever  we  teach,  is  carelessness,  in- 
y  accuracy,  forgetfulness,  and  slovenliness.  That  battle  has 
been  fought  and  won  with  diverse  weapons.  It  has,  for  in- 
stance, been  fought  with  Latin  grammar  before  now  and  won. 
I  say  that,  because  we  must  be  very  careful  to  guard  against 
the  notion  that  there  is  any  one  panacea  for  this  sort  of  thing. 
It  borders  on  quackery  to  say  that  elementary  physics  [and  I 
should  add,  elementary  mathematics]  will  cure  everything. 
The  personality  and  learning  of  the  teacher  are  everything 
here.  Nothing  else  matters  very  much.  That  ought  to  be  re- 
iterated by  every  educational  meeting  that  is  held." 

I  will  revert  to  this  question  of  the  personality  of  the 
teacher  under  another  head.  We  are  concerned  now  with  the 
benefits,  non-material  and  unprofessional,  yet  none  the  less 
real  and  practical,  that  are  derived  from  mathematical  study 
•by  the  majority  of  students.  '^  And  one  of  these  is,  as  we  have 
just  seen,  the  conquest  of  ''carelessness,  inaccuracy,  forget- 
fulness, and  slovenliness"  in  thinking  and  acting. 

USABLE   KNOWLEDGE  AND  POTENTIALIZING   KNOWLEDGE 

Let  US  assume,  for  the  moment,  however,  that  the  other  goal 
also  is  attainable,  —  namely,  the  acquisition  of  concrete,  usable 
mathematical  knowledge.  Let  us  admit  that  ten  per  cent  of 
our  high  school  graduates  actually  do  acquire  a  mathematical 
knowledge,  as  distinguished  from  a  mathematical  culture, 
which  they  may  be  expected  to  apply  to  physical  and  men- 
surational  problems;  let  us  admit,  too,  that  all  graduates  of 
technical  schools  and  colleges  possess  that  knowledge ;  the  ques- 
tion remains  how  long  do  they  retain  that  knowledge  and 

12 


whether  they  ever  in  the  majority  of  cases  possess  it  in  a  suffi- 
ciently powerful  and  ready  form  to  apply  it  universally  and 
successfully  in  practice.  I  personally  know  many  successful 
engineers,  but  I  know  very  few  who  know  mathematics  well 
and  who  can  use  it  as  a  powerful  auxiliary  tool  in  the  solution 
of  new  engineering  and  mechanical  problems.  Their  knowl- 
edge of  the  differential  and  integral  calculus  is  a  faint  dream 
of  their  collegiate  career.  They  are  devotees  of  elementary 
methods  and  rule-of-thumb  procedures  in  the  higher  domain. 
A  few  formulae,  a  few  tables  of  integrals  and  collections  of 
mechanical  rules  and  tabulated  calculations  are  their  chief 
stock-in-trade.  Their  sole  pabulum  is  what  I  might  call 
*' canned"  mathematics,  using  that  term  in  its  best  and  most 
esoteric  sense,  and  not  in  its  present  vulgar  and  figurative 
perversion,  —  that  capitalized  stock  of  mathematical  thought 
which  is  preserved  in  the  economy-jars  of  logarithmic  and 
other  tables  and  in  routine  formulae,  and  which  constitutes, 
in  this  capitalization  and  economization,  a  cardinal,  essential 
feature  of  mathematical  intellectual  activity.  This,  ninety-  ^ 
nine  out  of  every  one  hundred  engineers  will  admit.  But 
they  will  not  admit,  nevertheless,  that  they  have  derived  no 
^  profit  from  their  mathematical  training;  that  they  have  not  • 
preserved  from  it  that  unalysable  power  of  thinking  and  • 
visualizing  things  mathematically  and  geometrically  and  o:B 
pre-constructing  in  symbols  of  thought  and  pictures  of  the 
imagination  the  projects  which  afterwards  it  is  their  business 
to  create  in  space  with  physical  materials. 

Now,  what  I  contend  under  this  point  is,  that,  precisely  as 
the  engineer,  who  is  trained  for  mathematics  yet  usually  can-  ^ 
not  and  does  not  use  his  mathematics,  nevertheless  derives  in- 
estimable benefit  and  power  from  his  mathematical  education ;  ^ 
so  the  average  high  school  and  university  graduate  wTio  is 
never  expected  to  use  his  mathematical  knowledge  at  all,  in- 
variably acquires  from  his  mathematical  instruction  great 
moral  and  intellectual  impressions  and  potential  capacities 
of  all  sorts  which  will  redound  to  his  power  and  usefulness  in 
every  walk  of  life.  And,  in  making  this  a  cardinal  benefit 
derived  by  the  masses  from  mathematical  study,  I  am  not 

13 


slurring,  in  the  slightest,  the  great  material  benefits  of  mathe- 
matical knowledge  as  aif  auxiliary  engine  of  research,  control 

^nd  creation  in  every  profession.  I  merely  say  that  this  lat- 
ter, important  as  it  is,  constitutes  the  least  general  practical 
acquisition  from  a  mathematical  training;  while  the  former 
makes  up  in  all  cases,  some  nine-tenths  of  its  value.  For,  if 
this  were  not  so,  then  instruction  in  secondary  mathematics 
should  be  limited,  as  the  public  would  largely  limit  it,  to 

p  arithmetic  and  practical  mensuration.  And  we  should  thus 
save  at  least  one  year  of  our  time  for  the  pure  bread-and- 
butter  studies.  For,  how  otherwise  could  educators  justify 
this  economic  fact,  which  I  cannot  prove  statistically  but 
which  I  believe  to  be  true,  that  there  is  more  human  time  and 
energy  spent  on  writing  text-books  of  mathematics,  even  ap- 
plied mathematics,  and  on  teaching  mathematics,  than  is  ever 
after  employed  by  all  the  people  in  the  world  in  applying  it 
to  scientific  and  engineering  problems! 

AESTHETIC  CULTURE 

J  should  not  leave  this  aspect  of  our  subject  without  at 
least  briefly  referring  to  a  subtle  phase  of  purely  aesthetic 
culture  which  occupation  with  mathematical  studies  inevitably 
imparts,  —  although  this  consideration  sinks  behind  the  more 
practical  intellectual  benefits  I  have  above  referred  to.  There 
is  in  all  mathematical  research  a  genuinely  aesthetic  intel- 
lectual, as  distinguished  from  sensuous,  enjoyment,  issuing 
from  the  contemplation  of  the  unity,  symmetry,  and  dramatic 
movement  of  mathematical  creations,  which  is  observable  even 
in  the  logical  upbuilding  of  the  elementary  parts  of  alegbra  and 
geometry,  and  which  should  not  be  underrated  in  its  educa- 
tional effects.  This  intellectually  artistic  side  of  mathematical 
study  has  found  expression  in  all  great  thinkers  from  Pytha- 
goras to  Pascal,  Lagrange,  and  Kant,  and  has  had  its  praises 
sounded  by  many  lesser  and  later  writers  (Kummer,  Ilelm- 
holtz,  Boltzmann  and  Poincare).  But  I  must  abandon  this 
theme  for  some  more  pressing  topics.  I  have  omitted  in  this 
paper  all  special  discussion  of  the  logical  and  disciplinary  ad- 
vantages to  be  derived  from  mathematical  study,  as  also  the 
utilitarian  benefits  derived  from  its  pursuit.     These  aspects 

14 


of  the  subject  have  been  so  often  discussed  as  to  be  now 
commonplace  and  almost  self-evident. 

PEDxlGOGrCAL    COMPLICATIONS    AND    ECONOMY    OF    PRESENTATION 

The  considerations  adduced  above  lead  now  to  an  important 
practical  pedagogical  difficulty.  We  seem  to  be  seeking  in 
our  high  school  curricula  and  text-books  to  devise  a  universal 
pedagogical  machine  for  the  instruction  of  everybody  en 
masse  for  every  possible  end,  —  cooks,  dressmakers,  and 
scribes,  engineers  and  ''rope-stretchers,"  professional  "ad- 
mirable Crichtons, ' '  and  Jacks  of  all  intellectual  and  utilitari- 
an trades.  And  as  a  result  of  this  our  manuals  have  become 
a  veritable  mathematical  polyglot  and  Tohu-va-bohu.  A 
glance  at  the  problems  of  some  recent  admirable  algebraic  and 
geometrical  text-books  will  amply  show  this.  They  seek,  in 
their  problems,  to  cover  the  entire  universe  of  applied  knowl- 
edge, and  some  additional  domains  besides.  And  this  is  a 
gratifying  and  refreshing  symptom,  growing  out  of  a  genuine 
desire  to  modernize  and  vitalize  mathematical  instruction. 
But  we  forget  in  our  new-bom  practicalizing  zeal  that  all  this 
has  been  done  befoi  >  and  that  the  different  epochs  of  the 
long  development  of  mathematical  instruction  have  shown  the 
same  ideal  and  trend.  Each  period  has  had  its  practical  prob- 
lems, some  of  them  purely  intellectual,  but,  according  to  my 
view,  still  practical.  What  we  now  regard  in  the  older  text- 
books as  pure  rubbish  and  antiquated  intellectual  ' '  survivals ' ' 
were  once  mostly  real  problems.  But  to  show  this,  interesting 
as  it  is,  would  be  to  write  a  history  of  civilization,  for  which 
I  have  at  present  not  the  time.  I  merely  wish  to  call  to  your 
attention  the  fact  that  in  the  past  the  same  striving  for  mod- 
ernity and  yitalization  led,  as  it  will  likely  lead  now,  to  an 
almost  uncontrollable  emharras  des  richesses;  and  that  this 
superfluity  of  material  forced  the  old  textbook  writers,  from 
sheer  considerations  of  economy,  to  adopt  the  apparently 
harsh  and  arid,  abstract  method  of  presentation  which  we  now 
so  unanimously  condemn,  and  to  formulate  general  abstract 
problems  for  exercise,  instead  of  a  multitude  of  concrete  ap- 
plied problems.  For  economy  of  presentation  is  of  the  very 
essence  of  our  science  and  is  as  desirable  in  text-books  from  a 

15 


purely  practical  point  of  view  as  from  a  theoretic  one.  It 
was  doubtless  this  consideration  as  much  as  the  gibes  of  the 
Sophists  that  led  Euclid  to  the  formulation  of  his  Elements; 
and  any  one  who  doubts  this  view  has  but  to  compare  the 
beautiful,  lucid,  and  concrete,  but  interminably  prolix,  treat- 
ises of  the  great  Euler  in  the  18th  century  with  the  concise 
and  condensed  expositions  of  the  generation  that  followed 
him,  which  we  now  so  cordially  anathematize.  The  pedagogic 
pendulum  swings  back  and  forth  by  a  natural  intellectual  law. 
And  in  less  than  ten  years,  I  prophesy,  the  same  economic 
need  will  present  itself  in  our  American  text-book  world. 

ENDS  AND  MOTIVES 

Education,  even  mathematical  education,  is  a  class-problem, 
a  caste-problem,  an  individual  problem ;  and  what  we  need  here 
is  to  differentiate.  To  make  thinking  men  and  women  is  one 
thing,  to  make  pure  mathematicians  is  another,  and  to  make 
bookkeepers,  engineers,  mechanics,  etc.,  is  a  third.  Each  end 
is  legitimate  in  itself.  Each  requires  its  own  pedagogical  ma- 
chinery. But  in  the  average  text-book  and  in  the  average  cur- 
riculum of  the  public  school  it  is  sought  to  realize  all  these  ends 
at  once.  And  hence  results  a  necessary  confusion  and  conflict 
of  means  and  ends,  which  in  a  universal  system  of  education  it 
is  difficult  and,  it  may  be,  inexpedient,  to  disentangle.  But,  in 
undifferentiated  curricula  and  school-systems  such  as  ours,  we 
should  always  bear  this  in  mind,  in  our  discussions  and  striv- 
ings for  solutions.  What  part  and  amount  of  our  work  should 
be  devoted  to  cultivating  mathematics  as  a  science  in  itself,  as 
an  independent  body  of  natural  knowledge  having  its  own 
ends  and  methods ;  what  part,  to  it  as  a  purely  logical  and  cul- 
tural discipline;  what  part,  to  it  as  a  science  auxiliary  to 
physics,  mechanics,  surveying,  and  engineering,  or  as  a  body 
of  pure  useful  knowledge. 

These  points  of  view  should  be  considered  as  ends ;  but  they 
may  also  be  considered  in  the  role  of  their  efficiency  as  at- 
tractions or  stimuli  to  the  study  of  mathematics,  and  a  pro- 
portionately preponderant  share  accorded  to  each  of  them  ac- 
cording to  this  aspect.  And  here  the  practical  motives  appear 
to  be  predominant,  especially  in  the  applications  and  the  se- 

16 


lection  of  problems.  For  it  is  as  essential  in  pedagogics  as  in 
dietetics,  that  we  should  ''first  catch  our  hare  before  we  cook 
it,"  or,  as  a  writer  whom  I  have  before  quoted,  has  phrased 
it,  with  some  slight  approximation  to  profanity,  ''  that  what 
we  want  our  boys  and  girls  to  believe  is  that  mathematics  is 
indispensable  in  their  daily  life  and  not  something  they  will 
have  to  do  in  hell. ' '  ^  But  this  aspect  of  the  question  will  be 
adequately  discussed  by  the  gentlemen  that  are  to  follow  me.^ 
Practical  baits  are  legitimate,  but  they  are  not  the  end  of 
educational  psychology. 

PRACTICAL  MATHEMATICS  AND  PRACTICAL  PEDAGOGY 

1  will  now  indicate  the  general  character  and  trend  of  the 
so-called  practical  or  Perry  movement  in  the  tekching  of  math- 
ematics, which  constitutes  in  many  of  its  features  one  of  the 
most  stimulating  phases  of  recent  educational  thought.  Some 
little  discussion  will  probably  place  it  in  its  true  light  as  a 
branch  of  the  practical  psychology  of  teaching  rather  than  as 
a  new  idea  in  educational  science. 

"  What  is  really  meant  by  the  term  Practical  Mathema- 
tics? "  This  question  is  asked  by  the  speaker  in  the  London 
Conference  above  referred  to.  "I  am  sure  Prof.  Perry's 
meaning  has  been  misunderstood,"  he  says.  ''  For  instance, 
in  the  early  days  of  the  movement,  he  set  a  question  about  the 
relation  between  the  capacity  of  a  saucepan  and  its  price. 
Some  gentleman  who  took  no  interest  in  saucepans  (I  suppose 
he  was  a  bachelor),  took  great  exception  to  this  question,  and 
inferred  that  '  Practical  Mathematics  '  was  a  tin-pot  business. 

"  Now,  what  does  Practical  Mathematics  mean?  I  doubt 
if  Prof.  Perry  would  be  prepared  to  name  a  theorem  that  is 
certainly  not  '  practical. '  If  I  understand  him  rightly,  he 
means  the  term  to  apply  as  in  '  practical  politics. '  You  must 
consider  not  only  what  is  right  in  the  abstract,  but  what  is 
right  under  given  conditions.  The  fundamental  theorem  of 
Practical  Mathematics  is  thisi    There  is  not  only  the  math- 

1 ' '  Mathematical  Gazette, ' '  Londt)n,  Jan.,  1909,  p.  24. 

2  See  the  program  of  the  Mathematics  Section  of  the  Secondary  Con- 
ferences of  the  National  Educational  Association  for  1910,  Boston  meet- 
ing. 

17 


ematical  proposition,  but  there  is  the  boy  into  whose  head  that 
proposition  is  to  be  put,  and  the  spirit  in  which  the  boy  re- 
ceives the  proposition.  Prof.  Perry's  argument  is  that  the 
mathematician  is  apt  to  deal  with  the  wrong  half  of  the  prob- 
lem. He  worries  himself  about  Euclid  I.  47,  or  about  Fourier's 
Theorem.  There  is  nothing  wrong  with  them.  The  trouble  is 
with  the  pupil  who  is  to  receive  them. ' ' 

THE  PROBLEM  OF  PRACTICAL  PEDAGOGY 

This  last  remark  brings  us  to  the  human,  individual,  psy- 
chological problem  of  practical  pedagogy,  to  which  we  always 
have  to  revert  in  our  discussions,  with  which  all  general  solu- 
tions are  entangled,  from  which  no  text-book  can  save  us, — 
the  ever-recurring  problem  of  the  personality  of  the  individual 
pupil  and  the  personality  of  the  individual  teacher.  This 
problem  has  been  admirably  sununarized  by  the  great  German 
mathematician  Felix  Klein  in  the  following  manner.    He  says : 

''If  I  were  to  formulate  the  general  problem  of  pedagogy 
mathematically,  I  should  say  that  it  consisted  in  marshalling 
the  individual  qualities  and  capacities  of  the  teacher  and  his  n 
students  as  so  many  unknown  variables  and  in  seeking  the 
maximum  value  for  a  function  of  (1+n)  variables, 

F  (x^,x^,  ...x;^ ), 

under  given  collateral  conditions.  If  this  problem,  as  a  re- 
sult of  the  advances  made  by  psychology,  should  some  day  ad- 
mit of  direct  mathematical  solution,  then  from  that  day  on- 
ward practical  pedagogy  would  have  become  a  science.  Until 
then  it  must  refaain  an  art.''  {"  Deutsche  Math.-Yerein. 
Jahresber.,"  7,  1897-1898,  p.  133.) 

We  see,  thus,  that  no  text-book,  no  syllabus,  no  system  how- 
ever admirable,  is  ever  likely  to  free  us  from  the  necessity  of 
the  employment  of  individual  skill  and  tact  in  instruction. 
The  dominant  element  here  is,  and  has  always  been,  the  per- 
sonality, humanity,  and  range  of  scholarship  and  sympathy 
of  the  teacher. 

Mathematics  was  taught  as  well  one  hundred  and  fifty  years 
ago  as  it  is  today,  and  it  was  taught  as  poorly  one  hundred 
and  fifty  years  ago  as  it  is  today.     Wherever  there  exist  in 

18 


the  teacher's  mind  rich  associations,  a  broad  range  of  interests 
and  didactic  powers,  there  good  teaching  always  exists. 


OLD   AND    NEW   TEXT-BOOKS 

In  this  respect,  and  in  respect  to  text-books,  no  other  do- 
main can  admit  of  comparison  with  mathematics.  The  bio- 
logical and  descriptive  sciences,  chemistry  and  even  physics 
are  of  relatively  modern  development,  and  the  present  text- 
books, syllabi,  and  schemes  of  instruction  in  these  sciences 
are  enormously  superior  to  the  books  and  schemes  of  twenty- 
five,  fifty  or  seventy-five  years  ago,  even  where  these  exist; 
but  the  mathematics  which  is  now  taught  in  our  secondary 
schools  existed,  part  of  it  in  its  fullest  development,  as  geo- 
metry, 2100  years  ago,  and  part  of  it,  in  its  algebraic  devel- 
opment, two  hundred  years  ago.  In  the  years  between  1700 
and  1800  there  were  written  text-books  of  algebra  ^  in  which 
our  present  high-school  scholars  would  find  as  much  satisfac- 
tion and  from  which  they  would  derive  as  much  profit  by 
individual  study  as  from  some  of  the  books  published  within 
the  last  ten  years.  It  is  a  commonplace  of  educational  history 
that  geometry,  as  taught  in  our  schools,  is  in  its  didactic  form 
essentially  the  geometry  of  Euclid's  Elements;  and  I  make 
bold  to  say  that,  apart  from  the  supplying  of  practical  mo- 
tives in  teaching,  the  only  advance  which  has  been  made  in 
the  didactic  presentation  of  the  principles  of  geometry  in 
2000  years  is,  first^  the  reduction  of  the  verbiage  of  the  old  ele- 
ments and  the  economizing,  by  the  use  of  concise  language  and 
shorthand  symbols,  of  the  linguistic  form  in  which  the  proposi- 
tions were  presented  and  proved ;  and,  secondly,  the  introduc- 
tion into  the  books  of  practical  pedagogical,  typographical  and 
pictorial  devices  which  have  heightened  the  economy  of  sensual 
presentation  and  removed  all  physical  sense-obstacles  to  the 
comprehension  of  the  geometrical  relations,  —  an  enormous 
advance  in  itself  but  one  of  which  we  should  thoroughly 
understand  the  scope  and  which  we  should  not  overrate, 
recognizing  that  it  is  an  advance  which  we  owe  nearly  as  much 
to  the  development  of  the  arts  of  the  printer  and  the  illus- 
trator as  we  do  to  the  ingenuity  of  the  mathematical  author. 

1  By  Clairaut,  Euler,  Lagrange,  Laplace,  et  ceteris. 

19 


Our  science,  at  least  so  far  as  we  are  concerned  with  it  as 
secondary  teachers,  is  not  a  new  and  growing  science,  but  re- 
ceived its  full  development  centuries  ago;  so  much  so,  that 
one  of  the  greatest  mathematicians  of  the  18th  century,  La- 
grange, conscious  of  the  exhaustion  to  which  he  was  carrying 
his  mathematical  analysis,  once  remarked  that  it  would  not  be 
many  years  before,  for  the  purpose  at  least  of  pure  inquiry 
and  the  discovery  of  new  truth  along  the  old  lines,  professors 
of  mathematics  would  be  as  rare  at  the  universities  as  pro- 
fessors of  Arabic.^  This  fortunate  fact  should  be  remembered 
in  all  our  discussions.  No  mathematical  author  is  now  per- 
plexed with  doubts  as  to  the  new  subject-matter  which  he 
should  introduce  into  his  text-book  as  is  the  physicist  for  ex- 
ample when  he  comes  to  write  about  ions.  This  being  the  fact, 
our  task,  in  my  opinion,  in  both  selecting  and  presenting  the 
mathematical  subject-matter  in  text-books  is  limited  almost 
wholly  to  removing  the  sensual  physical  obstacles  to  the  intel- 
lectual comprehension  of  geometrical  and  algehraic  truths. 
It  is  a  question  of  pedagogic  economy.  And  how  magnificently 
this  is  being  done  by  color  devices,  photographs,  models,  sys- 
tematic and  universal  economic  notations  and  designations  in 
both  our  geometry  and  algebra  text-books  is  known  to  every 
one  who  has  seen  some  of  the  recent  beautiful  productions  of 
our  American  text-book  press. 

HISTORY  OP  THE  PRACTICAL  MOVEMENT 

This  beauty,  precision  and  economy  of  form  which  our  mod- 
ern text-books  are  taking  on,  and  this  supplying  of  practical 
motives,  have  been  the  outgrowth  of  a  long  development,  which 
may  be  unknown  to  some  of  you  and  of  which  the  recent  con- 
crete practical  methods  of  teaching  mathematics  are  themselves 
the  outcome.  The  movement  is  not  so  new  as  one  would  infer 
from  the  prefaces  to  the  American  text-books  of  the  last  fif- 
teen years.  The  French  Revolution  gave  the  first  universal 
impulse  to  the  change,  and  in  the  Normal  and  Polytechnic 

1  For  a  refreshing,  but  drastic  and  warped,  characterization  of  the 
history  of  mathematical  instruction  and  text-books  see  Eugen  Diihring's 
masterful  "Geschichte  der  Mechanik,'^  Leipsic,  Fues's  Verlag.  Diihr- 
ing's  work  is  little  known  in  this  country. 

20 


schools  of  Paris  over  one  hundred  years  ago  graphs,  for  ex- 
ample, were  used  by  the  greatest  mathematicians  of  Europe  to 
illustrate  the  principles  of  algebra  —  although  one  would  think 
from  some  recent  writers  that  this  device  had  been  the  pro- 
duct, if  not  of  their  own  brains,  at  least  of  very  recent  years. 

But  the  great  educational  reformers  of  Germany  hastened 
this  process  in  the  elementary  schools.  Trendelnburg  was 
prominent  in  attacking  the  blind  deductive  form  in  which  the 
Euclidean  geometry  was  taught  in  the  schools  and  to  ask  for 
a  more  concrete  treatment.  Schopenhauer's  works  are  full  of 
onslaughts  on  the  sterile,  formal,  deductive  methods  of  teach- 
ing geometry,  among  which  the  one  making  fun  of  Euclid's 
proof  of  the  Pythagorean  proposition,  which  Schopenhauer 
called  the  "  mouse-trap  proof,"  is  the  most  classical.  Study- 
ing geometry  by  Euclid's  method,  Schopenhauer  said,  "  was 
like  cutting  off  one 's  legs  in  order  to  walk  on  crutches. ' '  ^ 

But  it  was  the  influence  of  Herbart  that  did  most  toward 
giving  geometrical  teaching  its  right  bearings  and  toward 
starting  the  movement  which  has  resulted  in  the  methods  of 
such  men  as  Perry  and  of  scores  of  other  equally  effective 
teachers  and  schools  in  our  own  country  whose  names  do  not 
happen  to  be  known.  ' '  This  is  the  ideal  that  seeks  simply  to 
instill  into  the  mind  fruitful  ideas,  ideas  which  find  a  con- 
genial soil  in  the  student's  existing  knowledge  and  interests, 
and  which  from  a  firm  belief  in  the  fertility  of  ideas  has  em- 
phasized the  value  of  geometrical  knowledge  as  opposed  to  the 
study  of  logical  form.  Let  the  boy, ' '  it  says,^  '  *  be  thoroughly 
at  home  with  a  new  fact  or  property  before  he  begins  to  apply 
formal  logic  to  it.  To  attain  this  familiarity,  do  not  reject  at 
any  stage  the  help  of  [physics,  mechanics  and]  experiment, 
and  the  recourse  to  common  objects  and  experience.  Geomet- 
rical experiment  may  use  models,  frameworks,  machines;  but 
there  is  a  limit  to  the  amount  of  apparatus  that  is  convenient. 
We  rely,  therefore,  in  the  main,  upon  figures :  freehand  sketch- 

1 '  *  W.  als  W.  u.  V.  ^ '  I.  1.  15.  Eousseau,  '  *  Confessions, ' '  has  a  similar 
remark. 

2  Abstracted  from  a  paper  in  the  ' '  Mathematical  Gazette, ' '  London, 
March,  1910.  This  excellent  little  magazine  should  be  accessible  to  all 
secondary  mathematics  teachers. 

21 


es,  where  a  sketch  will  reveal  the  fact  that  we  are  looking  for ; 
accurate  figures,  where  eye  and  hand  alone  are  not  clever 
enough.  Hence  the  amalgamation  of  geometrical  drawing  with 
geometrical  theory,  subjects  once  divorced,  to  the  great  loss 
of  both." 

THE    HUMAN    BRAIN    ITSELF    A    MATHEMATICAL    LABORATORY 

I  will  stop  here  a  moment  to  consider  how  far  experiment, 
physical  illustrations,  and  models  may  be  carried  in  the  teach- 
ing of  geometry.  This  tendency  has  been  greatly  exaggerated 
and  overdone  in  some  of  our  recent  literature.  I  have  seen 
in  recent  current  American  educational  magazines  several  sug- 
gestions for  the  physical  illustration  of  theorems  of  algebra 
and  geometry  which  remind  one  of  the  proverbial  procedure 
of  using  a  steam-hammer  to  crack  a  nut,  procedures  which  are 
altogether  unnecessary  and  a  sheer  waste  of  time.  Practical 
and  laboratory  methods  in  mathematics  and  mechanics  have 
their  limited  sphere  of  application ;  for  it  should  be  considered 
in  this  connection,  and  with  special  regard  to  our  own  science, 
that  the  human  brain  for  our  purpose  is  itself  a  store-house 
and  laboratory  of  formal  thought,  where  nearly  all  the  ex- 
periences are  collected  that  are  needed  for  experiments  in  al- 
gebra and  geometry,  and  which  can  be  here  conducted  with 
far  more  dispatch  and  ease  than  in  a  special  laboratory  with 
levers  and  scale-pans.  The  human  body  and  brain  are  a  micro- 
cosm, a  compact  bundle  of  well-ordered  space  and  mechanical 
experiences,  which  are  always  ready  at  hand.  The  chalk  and 
the  blackboard  in  our  science  are  mightier  than  all  the  tin-can 
junketry  of  the  physical  laboratory. 

METHODIC  AND  GENETIC  METHODS 

Several  attempts  also  were  made  in  the  early  part  and  mid- 
dle of  the  19th  century  to  write  treatises  in  which  the  subject- 
matter  and  contents  of  geometry,  entirely  apart  from  its  form, 
should  be  developed  genetically,  as  would  mechanics  or  elec- 
tricity, by  the  use  of  the  principles  of  motion  and  continuity, 
and  which  discarded  entirely  the  old  arid  and  sterile  deduct- 
ive form.  But  let  me  remark,  parenthetically,  that  the  re- 
jection of  the  old  deductive  form  of  presentation  is  not  neces- 
sarily a  rejection  of  the  deductive  form  of  reasoning.    We  are 

22 


not  losing  the  value  of  the  training  in  formal  discipline  when 
we  study  the  contents  of  mathematics,  purely  for  their  con- 
tents '  sake.  This  is  what  I  mean  when  I  say  that  the  two  ideals 
which  most  people  seem  to  separate  in  recommending  the 
study  of  mathematics, —  namely,  the  formal  ideal  and  the 
utilitarian  ideal, —  are  not  necessarily  contradictory  and  not 
necessarily  exclusive.  Some  of  these  genetic  attempts  ^  have 
not  yet  been  fully  exploited  in  our  American  secondary  text- 
books, which  still  halt  between  the  two  ideals,  and,  like  the 
mediaeval  donkey  between  the  two  bales  of  hay,  find  intellectual 
starvation  while  hesitating  to  decide  which  sustenance  to  take. 
But,  rejecting  both  the  formal  and  the  genetic  methods,  and 
almost  spurning  mathematics  as  an  independent  science,  the 
Perry  movement  takes  the  ultra-utilitarian  bull  by  the  horns, 
and  seeks  to  convert  mathematics  into  a  mere  auxiliary  en- 
gine of  physical  and  mensurational  practice.  This  ideal  is 
that  the  boy  should  possess  such  a  ' '  command  of  mathematical 
methods  that  he  can  apply  them  in  any  new  problems  with 
ease  and  certainty.  There  should  be  no  separate  examinations 
in  geometry,  algebra,  trigonometry,  and  mechanics.  They  are 
all  one  subject.^  Geometry  should  not  be  a  study  of  logic,  but 
should  be  a  study  of  mathematical  matter  and  method  with 
experiment  and  common-sense  reasoning.  All  parts  of  the 
elementary  mathematics  ought  to  be  taught  along  with  and 
through  science  and  by  the  same  master.  [A  rather  difficult 
requirement  for  American  schools.]  Common-sense  explana- 
tion, accompanying  experiiiiBnt,  should  be  the  procedure.  Our 
great  difficulty  is  our  being  out  of  touch  with  our  students. 
We  must  bring  the  teachers  to  see  down  to  the  level  of  the 
pupils  and  to  think  what  the  world  needs.  The  traditional 
mode  of  teaching  mathematics  does  not  provide  adequate  mo- 
tives for  the  work  from  the  point  of  view  of  the  boy,"  we 
read,^  and  the  Perry  reform  is  to  take  measures  that  such 
motives  shall  be  supplied.     Mathematics  becomes  simply  an 

1  For  example,  that  of  Snell,  in  a  now  almost  unknown  book,  ' '  Lehr- 
buch  der  Geometric, "  Leipsic,  1869. 

2  Compare,  as  a  variant  of  this  plan,  books  like  Holtzmiiller  's  ' '  Meth- 
odisches  Lehrbueh  der  Elementar-Mathematik, "  Leipsic,  Teubner. 

3  *' Mathematical  Gazette,'^  March,  1910. 

23 


auxiliary  instrument  for  dealing  with  problems  of  a  mechan- 
ical, physical,  and  mensurational  character.  Formal  mathe- 
matical technique  is  a  secondary  consideration.  The  successful 
use  of  the  instrument,  no  matter  how  clumsy  the  use,  is  the 
main  end. 

I  am  tempted  here  to  present  some  considerations  on  the 
relative  value  of  the  Perry  method  of  teaching  mathematics 
and  the  so-called  ''  methodic  "  methods  of  such  foreign  writ- 
ers as  Holtzmiiller,  as  also  of  the  pure  ' ' genetic"  method,  where 
the  entire  subject-matter  of  mathematics  is  treated  as  an  organ- 
ic body  of  knowledge  which  receives  didactic  presentation  not 
in  the  separate,  sundered  form  of  arithmetic,  algebra,  geo- 
metry, trigonometry,  etc.,  but  as  a  unitary  system  of  con- 
solidated facts,  concepts,  and  ideas.  This  last  method  of  pre- 
sentation, as  supplemented  by  the  Perry  procedures,  is  com- 
ing generally  to  the  fore  in  our  text-book  literature  and  un- 
doubtedly constitutes  the  ultimate  goal  toward  which  math- 
ematical didactics  strive.  But  I  can  do  nothing  further  here 
than  to  supply  intimations.  We  are  concerned  merely  with 
the  philosophy  and  history  which  enter  into  these  various  con- 
ceptions, and  I  shall  conclude  merely  with  some  remarks  on 
how  they  have  arisen. 

SUMMARY 

All  the  varying  arguments,  all  the  contradictory  points  of 
view  above  intimated,  all  this  emphasis  now  of  one  side  and 
now  of  the  other  side  of  the  value  of  mathematical  studies,  are 
due  to  a  lack  of  historical  and  philosophical  comprehension. 
In  its  origin,  mathematics  was  developed  largely  as  an  instru- 
ment for  physical  or  mensurational  research.  After  it  was 
developed,  and  the  logical,  genetic  interdependence  of  its 
truths  and  results  was  discovered,  it  became  naturally  an  ob- 
ject of  independent  development  in  itself,  and  assumed  neces- 
sarily an  artificial,  logical  and  purely  conceptual  character. 
It  created  a  professional  caste,  like  the  grammarians  of  India, 
which  cultivated  it  for  its  own  sake ;  it  became  an  end  in  itself ; 
it  was  systematized,  its  principles  economized  and  minimized, 
and  its  entire  body  of  truth  transformed  and  etherealized 
into  a  shadowy  framework  of  pure  logical  forms;  it  lost  its 

24 


foothold  in  reality  and  in  the  empirical  soil  from  which  it  had 
sprung,  became  barren  and  sterile ;  it  ceased  to  produce  fruit- 
ful and  creative  investigators,  and  was  ultimately  rejected  in 
this  form  by  practical  minds.  Hence  the  contradictory  opin- 
ions and  utterances  on  the  value  of  the  two  phases  of  its  de- 
velopment as  an  educational  discipline ;  and  hence  the  varying 
hyper-emphasis  that  the  one  or  the  other  aspect  of  it  receives, 
according  to  the  changing  needs  and  ideals  of  each  historical 
epoch.  The  truth,  to  which  we  must  hold,  lies  in  neither;  it 
must  be  sought  in  an  amalgamation  of  the  two:  mathematics 
arose  from  a  physical,  empirical  soil ;  it  reverts  in  its  applica- 
tions to  that  soil ;  but  in  the  transition  it  has  passed  through 
a  purely  intellectual  domain,  in  which  it  has  suffered  a  trans- 
formation that  is  of  its  very  essence.  And  the  study  of  its 
subject-matter,  its  logical  and  genetic  development  in  this 
transformation  is  just  as  important  an  educational  discipline 
for  the  intellectual  needs  of  life  as  the  study  of  its  applications 
is  for  the  material  needs  of  life.  And  the  former  objects  are, 
I  think,  more  easily  and  more  frequently  obtained  in  practice 
than  the  latter. 

Two  antithetic  quotations  in  closing :  Hankel,  a  Tvell-known 
German  historian  of  mathematics,  says:  "  Mathematics  will 
never  find  its  adequate  appreciation  among  the  general  pub- 
lic until  more  than  its  A  B  C's  are  taught  in  the  schools,  and 
until  the  unfortunate  opinion  has  been  removed  that  its  sole 
object  in  instruction  is  to  impart  formal  culture  to  the  mind. 
Mathematics  finds  its  goal  and  purpose  in  its  contents:  its 
form  is  a  secondary  consideration  and  not  necessarily  that 
which  it  has  come  to  be  historically  from  the  fact  that  it  first 
took  fixed  shape  under  the  influence  of  the  Grecian  logic. 
There  is  no  more  reason  for  studying  mathematics  for  its 
formal  culture  than  there  is  for  studying  history  to  strengthen 
the  memory. ^^  ^ 

This  is  the  utterance  of  the  extreme  technical  party.  It  is 
attractive  and  forcible,  even  necessary  in  a  certain  stage  of 
the  development  of  educational  practice,  but  does  not  contain 
all  the  truth.    W.  Kingdon  Clifford  has  expressed  the  opposite 

1  Hermann  Hankel,  *  *  Die  Entwickelung  der  Mathematik  in  den  letzten 
Jahrhunderten, "     Tiibingen,   1869. 

25 


view,  which  holds  to  the  more  formal  teaching  of  mathematics 
as  a  branch  of  pure  scientific  culture. 

"  It  seems  to  me,"  he  says,  '*  that  the  difference  between 
scientific  and  merely  technical  thought  ....  is  just  this: 
Both  of  them  make  use  of  experience  to  direct  human  action ; 
but  while  technical  thought  or  skill  enables  a  man  to  deal  with 
the  same  circumstances  that  he  has  met  with  before,  scientific 
thought  enables  him  to  deal  with  different  circumstances  that 
he  has  never  met  with  before. ' ' 

I  will  conclude  with  these  two  quotations,  leaving,  for  your 
consideration  simply,  the  plenitude  of  implications  they  in- 
volve. I  have  endeavored,  rather  by  way  of  intimation  than 
by  detail,  to  present  a  few  central  thoughts  that  this  vast  sub- 
ject adumbrates,  in  the  hope  that  their  mere  suggestion  will 
lead  to  some  light  and  clarification.    In  magnis  voluisse  sat  est. 


1  "Lectures  and  Essays,"  Vol.  I,  p.  128. 

26 


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